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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1536 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1536.1 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
bnj1536.2 | ⊢ (𝜑 → 𝐺 Fn 𝐴) |
bnj1536.3 | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
bnj1536.4 | ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)) |
Ref | Expression |
---|---|
bnj1536 | ⊢ (𝜑 → (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1536.4 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)) | |
2 | bnj1536.1 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
3 | bnj1536.2 | . . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴) | |
4 | bnj1536.3 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
5 | fvreseq 6227 | . . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐵 ⊆ 𝐴) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))) | |
6 | 2, 3, 4, 5 | syl21anc 1317 | . 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))) |
7 | 1, 6 | mpbird 246 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∀wral 2896 ⊆ wss 3540 ↾ cres 5040 Fn wfn 5799 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 |
This theorem is referenced by: bnj1523 30393 |
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